Introduction to variables · EduSite Academy

▶ What is a variable?

Variables in math are symbols, often letters, that represent different values in various situations. They help us understand and solve problems with changing values. For example, when calculating total earnings at a job with an hourly wage plus tips, a variable can represent the fluctuating tips, making it easier to determine total income.

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▶ Why aren't we using the multiplication sign?

In algebra, representing multiplication with variables can be tricky due to the similarity between the variable "x" and the multiplication symbol. To avoid confusion, use alternative methods like 2⋅x, 2(x), or 2x. Practice evaluating expressions by substituting given values for variables and following the order of operations.

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▶ Creativity break: Why is creativity important in STEM jobs?

Experts share why creativity plays an important role in STEM jobs.

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▶ Evaluating an expression with one variable

In this math lesson, we explore the expression 5 t plus 3 to calculate the cost of participating in a hospital raffle fundraiser. By evaluating the expression for different values of t, the number of tickets purchased, we find the costs for buying 1, 8, and 10 tickets, which are 8, 43, and 53 dollars, respectively.

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📄 Evaluating expressions with one variable

##How to evaluate an expression with one variable

Let's say we want to evaluate the expression $a + 4$. Well, first we need to know the value of the variable $a$. For example, to evaluate the expression when $\blueD {a = 1}$, we just replace $\blueD a$ with $\blueD 1$:

$\begin{align}
&\blueD a + 4
\\
=&\blueD1 + 4~~~~~~~~\gray{\text{Replace }\blueD{a} \text{ with } \blueD{1}\text{.}}
\\
=&5
\end{align}$

So, the expression $a + 4$ equals $5$ when $a = 1$.

We can just as easily evaluate $a + 4$ when $\blueD {a = 5}$:

$\begin{align}
&\blueD a + 4
\\
=&\blueD5 + 4~~~~~~~~\gray{\text{Replace }\blueD{a} \text{ with } \blueD{5}\text{.}}
\\
=&9
\end{align}$

So, the expression $a + 4$ equals $9$ when $a = 5$.

##Evaluating an expression with multiplication

You might be asked to "Evaluate $3x$ when $x = 5$."

Notice how the number $3$ is right next to the variable $x$ in the expression $3x$. This means "$3$ times $x$". The reason we do this is because the old way of showing multiplication with the symbol $\times$ looks confusingly similar to the variable $x$.

Okay, so now let's solve the problem:

$\begin{align}
&3\blueD x
\\
=& 3 \cdot \blueD5~~~~~~~~\text{Replace }\blueD{x} \text{ with } \blueD{5}\text{.}
\\
=&15
\end{align}$

So, the expression $3x$ equals $15$ when $x = 5$.

New ways to show multiplication

Hold on a second! Did you notice that we wrote "$3$ times $\blueD 5$" as $3 \cdot \blueD 5$ instead of as $3 \times \blueD 5$? Using a dot instead of the symbol $\times$ is another new way of showing multiplication:

$3 \cdot \blueD 5 = 15$

Parentheses can also be used to show multiplication:

$3(\blueD 5) = 15$

Let's summarize the new ways of showing multiplication that we learned.

| Old way | New way

| :-: | :-:
With a variable | $3 \times x$ | $3x$
Without variable | $3 \times 5$ | $3\cdot 5$ or $3(5)$

##Evaluating expressions where order of operations matter

For more complex expressions, we'll have to be sure to pay close attention to order of operations. Let's take a look at an example:

Evaluate $5 + 3e$ when $\blueD{e=4}$.

$\begin{align}
&5+3\blueD e
\\
=&5 + 3 \cdot \blueD 4~~~~~~~~\gray{\text{Replace }\blueD{e} \text{ with } \blueD{4}\text{.}}
\\
=&5 + 12 ~~~~~~~~\text{\gray{Multiply first (order of operations)}}
\\
=&17
\end{align}$

So, the expression $5 + 3e$ equals $17$ when $e = 4$.

Notice how we had to be careful to think about order of operations when evaluating. A common wrong answer is $\redD{32}$, which comes from first adding $5$ and $3$ to get $8$ then multiplying $8$ by $4$ to get $\redD{32}$.

##Let's practice!

[[☃ graded-group-set 1]]

##Challenge problems

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